Donald J. Trump during a news conference in Washington on Monday.Credit Doug Mills/The New York Times

Donald J. Trump and Senator Ted Cruz of Texas traded insults on Easter Sunday morning over recent smears against their wives, while Mr. Trump ruled out creating internment camps for American Muslims and said he would study a proposal to allow delegates to bring guns to the Republican National Convention.

Mr. Trump, the leading Republican candidate, categorically denied that he or anyone from his campaign played a role in a National Enquirer report on Friday suggesting that Mr. Cruz may have had extra-marital affairs. The report by the magazine came after Mr. Trump threatened to “spill the beans on” Mr. Cruz’s wife, Heidi, a cryptic remark that Mr. Trump elevated on Sunday into a dark insinuation.

“There are things about Heidi that I don’t want to talk about,” Mr. Trump said on ABC’s “This Week.” “You could look, but I don’t want to talk about them.”

Mr. Cruz, appearing on “Fox News Sunday,” repeated his contention that his rival’s fingerprints were on the report because it included a comment from a former Trump political adviser, Roger Stone, and because Mr. Trump is friends with the owner of the Enquirer. He called the report “garbage” and said it was “disgusting to see a candidate attacking the spouse of another” – while at the same time denying that he had anything to do with a mailing by a “Super PAC” featuring a mostly unclothed Melania Trump, Mr. Trump’s wife and a former model, that circulated in Utah before Tuesday’s primary there.

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Senator Ted Cruz of Texas during a news conference at 3 West Club in New York on Wednesday.Credit Hilary Swift for The New York Times

The warfare over spouses represents an attempt by Mr. Trump to get under Mr. Cruz’s skin, just like he did, effectively, with former rivals like Jeb Bush and Ben Carson. Mr. Trump also continues to use social media like Twitter to focus the race on personality and language rather than policy: He denied disparaging Mrs. Cruz by posting an unflattering photo of her on his Twitter page by noting that it was simply a retweet of another user’s post. He added that he would not use Twitter “very much as president” compared to his current tendency to send flurries of late-night tweets.

On “Fox News Sunday,” Mr. Cruz argued that Mr. Trump had a habit of turning to personal attacks “when he gets scared, particularly on foreign policy,” and said he was doing so now because he was “out of his depth” with recent comments like suggesting that United States pull back from NATO unless other nations pay more of the alliance’s costs. Mr. Trump repeated that view on Sunday and called NATO “obsolete” in the fight against terrorism.

Mr. Trump was also asked if he would rule out internment camps for American Muslims – an idea he has never proposed, though he has argued for a temporary ban on Muslims entering the country if they are not American citizens.

“I would rule it out, but we would have to be very vigilant,” Mr. Trump said. “If we’re not very, very strong and very, very smart, we have a big, big problem coming up. We’ve already had the problem. Check out the World Trade Center, check out the Pentagon. We’ve already had the problem.”

Mr. Trump declined to take a position on a petition to allow guns at the Republican convention, contending that delegates at the high-security event risk being “sitting ducks, utterly helpless against evil-doers and criminals.” Mr. Trump said he had not seen the petition, which is aimed at candidates like himself, and that he would have to review it before commenting, although he noted he was a “very, very strong person for the Second Amendment.”

The third Republican candidate, Gov. John Kasich of Ohio, who is running far behind his rivals in the race for the party’s nomination, expressed new hesitancy on Sunday to support Mr. Trump if he becomes the nominee.

“We’re going to look at it every single day, and we’ll see what happens,” he said on NBC’s “Meet the Press.” “We’ve got a long way to go. I don’t want to project that he’s going to be the nominee. I don’t think he will be. And if he is – we’ll have to, I will review it every day.” Pushed by moderator Chuck Todd on whether he would back Mr. Trump, Mr. Kasich said, “I said what I said, Chuck, and I’m done talking about this subject.”

In the Democratic race, Senator Bernie Sanders of Vermont appeared on three Sunday shows after his sweep of caucuses in Alaska, Hawaii, and Washington on Saturday. On CNN’s “State of the Union,” he said those and other recent “landslide victories” gave him “a path toward victory” in his race against Hillary Clinton to accumulate the 2,383 delegates needed to clinch the Democratic nomination.

Mrs. Clinton holds a strong delegate lead over Mr. Sanders, but after Saturday’s caucuses, it shrunk to about 265 pledged delegates from about 300. Mrs. Clinton’s lead remains all but insurmountable because both candidates will continue accruing delegates through the final primaries in June; delegates are awarded based on each Democrat’s proportion of the total vote. Mr. Sanders would need to win extremely large landslides in California, New Jersey, New York, Pennsylvania, and other states to overtake Mrs. Clinton in pledged delegates, which even some Sanders advisers say is unlikely.

On CNN, Mr. Sanders alluded to his best-case scenario: Winning enough pledged delegates to stop Mrs. Clinton from reaching the 2,383 magic number. If that happens, Democratic Party leaders and officials who are known as super-delegates would decide the race, as they did in 2008 in the nomination fight between Mrs. Clinton and Barack Obama.

In the current race a majority of super-delegates have sided with Mrs. Clinton, but they can change their minds at any time. Mr. Sanders said he would try to win them over with polls showing he would beat Mr. Trump by larger margins than Mrs. Clinton, and with arguments that super-delegates should back him if he won their states’ primaries and caucuses.

“A lot of these super-delegates may rethink their position with Secretary Clinton,” Mr. Sanders said. “A lot of them have not yet declared, and then you’ve got super-delegates who are in states where we win by 40 or 50 points. I think their own constituents are going to say to them, Hey, why don’t you support the people of our state – vote for Sanders.”

CNN’s Jake Tapper also asked Mr. Sanders about a recent news report suggesting that the senator would only support Mrs. Clinton as the Democratic nominee if she endorsed five of his policy priorities, like a single-payer health care system.

“No, I don’t think we’ve ever framed things in that sense,” Mr. Sanders said before focusing again on his own determination to win the nomination.

Mrs. Clinton had no media interviews or campaign events scheduled for Sunday.

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